He’s 90, won’t sign a POA, doesn’t want to pay for a sitter, nor nursing home. He’s feeble but wants to be waited on hand and foot then complains about everything and everyone. He constantly claims he’s dying and tells my sister and I “ it’s an emergency take me to ER; I don’t feel good.”
Sadly, you have no "standing" to place your dad.
His health will come to a crisis point, he will fall, or fall ill and end up in the hospital.
Once there, you will talk to the discharge planners and tell them that he lives alone, won't cooperate with getting help and that you are not able or willing to provide hands on care.
They will find placement for him and the NH he is admitted to may well file for guardianship.
There is no faster route to a nursing home bed than lack of cooperation with folks who have your best interests at heart.
Consider telling your dad that the hard choice is between having folks who love you find a placement, or having the state placing him in the first available bed.
His care is his problem, not yours.
It will feel very hard at first, then it will get much better once he is somewhere appropriate and social for him, and you and your sister can have peace of mind and your lives back.
Currently these sisters are enabling his behavior.
They need to make it clear to social services they cannot accept him back into their care if he is living with them; if they are living with HIM, they need to move.
Meanwhile, do know that if you are living with him, or he with you, then you are enabling this behavior. Do not accept him back into your car after hospitalization. He is uncooperative and you will not change his mind.
He is 90. Seems he is determined to pass in his own home.
The question now is do you wish to live with him and allow that, or are you unwilling to do so.